On open maps of Borel sets
نویسنده
چکیده
We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers P not Gδ · Fσ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ P of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. The answer to the question as stated is negative if X is Baire but Y is not Baire. Introduction. In 1934 Hausdorff proved [3; 2, 4.5.14] that if f : X → Y is an open map from a completely metrizable space X onto a metrizable Y , then Y is also completely metrizable. Thus, open maps preserve the class Gδ of Borel sets. L. V. Keldysh proved [5, Th. 1] that this result is not true for Borel sets of higher class, namely, that there is a Borel set X ⊂ P of the first category for which there is an open map f : X → Y onto an arbitrary analytic set Y ⊂ P (see Theorem 1). In connection with this result a question was raised whether an analogous theorem holds for Baire spaces. It is clear that if f : X → Y is an open map and O ⊂ Y is an open (nonempty) set of the first category, so is f−1(O). Hence, open maps preserve the property of being a Baire space. Let X0 ⊂ P be an analytic set such that P \ X0 does not contain a copy of the Cantor set C. It is not hard to see that X0 is a Baire space. Keldysh remarked that if Y satisfies the following condition: (i) Y ⊂ P is an analytic set such that M \ Y contains a copy of the Cantor set C for every Gδ-set M ⊃ Y , then X0 cannot be mapped onto Y by an open map [5]. Note that every Borel (non-Gδ) set Y ⊂ P (and analytic set Y = X0 × P in which every Gδ-subspace is Baire [12, Theorem 4]) satisfies the condition (i). 1991 Mathematics Subject Classification: 54H05, 54C10, 54C20, 03E15.
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